Measures of Central Tendency
Measures of Central Tendency
Descriptive statistics that offer information on where the scores in a data set tend to cluster.
Examples: Mean , Median , Mode .
Describes where the majority of scores within a data set reside.
Often summarized by asking, “Where is the middle?”
The Mean
The arithmetic average of a variable within a set of data.
\[\bar{x} = \frac{\sum x}{n}\]
\(\bar{x}\) = Mean
\(\sum\) = Summation
\(x\) = Values in a column
\(n\) = Sample size
A Mean Example
We have five college student’s test scores on a particular exam:
grades
student_1 85
student_2 90
student_3 75
student_4 86
student_5 91
\[\bar{x} = \frac{(85 + 90 + 75 + 86 +91)}{5}\]
\[\bar{x} = ?\]
Odd Number (Method One)
We only need one step for this!
\[Median = \frac {n + 1}{2} \]
\(n\) = Sample size
Even Number (Method Two)
LM and UM are numbers within the column.
Step 1
\[Lower Middle (LM) = \frac {n}{2} \]
Step 2
\[Upper Middle (UM) = \frac {n}{2} +1 \]
Step 3
\[Median = \frac {LM + UM}{2}\]
Method One Example
We have nine college student’s test scores on a particular exam:
grades
student_1 85
student_2 90
student_3 75
student_4 86
student_5 62
student_6 79
student_7 96
\[Median = \frac {? + 1}{2} \]
Have them work this out but if they forget to put in order, let them figure it out on their own but slightly lead them to it.
Method Two Example
We have four college student’s test scores on a particular exam:
grades
student_1 85
student_2 90
student_3 75
student_4 86
\[(LM) = \frac {?}{2} \] \[(UM) = \frac {?}{2} +1 \] \[ Median = \frac {(? + ?)}{2}\]
The Mode
The most frequent number in a set of scores or column .
Simply sort your values in ascending order, then count and compare (Hint: Look for numbers that repeat) .
A Mode Example
We have seven college student’s test scores on a particular exam:
grades
student_1 85
student_2 90
student_3 75
student_4 85
student_5 62
student_6 79
student_7 96
\[ Mode = ?\]
Normal Distribution
A set of scores that cluster in the center and tapers off to the left and right sides of the number line.
Outliers
Extreme values that pull the distribution which leads to a positive or negative skew.
Positive Skew
A clustering of scores in a distribution with some large scores pulled (or skewed ) toward the positive side of the x-axis
This occurs because of large scores (or outliers) along x-axis.
For example, the 150 - 200 number of sunspots.
Negative Skew
A clustering of scores in a distribution with some small scores that pulled (or skewed )the towards the negative side of the x-axis.
This occurs because of small scores (or outliers) along x-axis.
For example, the lower scores (e.g., 30) on the zoology test score.
Imagine on the other side (the left) of the 0 on the x-axis there are negative numbers.
Mnemonic Devices
These are memory techniques that aid in memory retention and retrieval.
Dates back to the early Greeks.
Helps transition short term to long term memory more quickly.
Skew Mnemonic
When you hear the word “skew” think of laying on a beach and putting your feet together while looking at the horizon.
Your ri ght foot is the posi tive skew.
Your le ft foot in the ne gative skew.
Pop Quiz
Using the data below, calculate the mean, median, and mode.
Wins_Under_Scott_Frost
2018 4
2019 5
2020 3
2021 3
2022 4
Mean = ?
Median = ?
Mode = ?
Recap
Measure of central tendency
Deviation Score
The deviation score is the distance between the mean of a variable and any given raw score in that variable.
\[d_i = x_i - \bar{x}\]
\(d_i\) = deviation score
\(x_i\) = a given raw score (i.e., data point)
\(\bar{x}\) = the mean
Focus on the raw score here, make sure they understand it. Use the board
Deviation Score
This tells us two important things:
1.) How far the raw score is from the mean (\(\bar{x}\) )
2.) Whether the raw score is greater or less than the mean (\(\bar{x}\) )
Positive deviation scores represent raw scores that greater than the mean.
Negative deviation scores represent raw scores that less than the mean.
Deviation Score Example
1.) Calculate the (\(\bar{x}\) )
2.) Calculate the \(d_i = x_i - \bar{x}\) for each student
grades
student_1 85
student_2 90
student_3 75
student_4 85
student_5 62
student_6 79
student_7 96
\(\bar{x}\) = 81.71
\(d_i\) = 3.28, 8.28, -6.71, 3.28, -19.71, -2.71, 14.28
Reiterate this again:
Positive deviation scores represent raw scores that greater than the mean.
Negative deviation scores represent raw scores that less than the mean.
Pop quiz
You are a researcher at a think tank and you are about to present new findings regarding the DARE program. Your data has a (\(\bar{x}\) ) of 45, a median that is 35, what type of skew will your sample have? And what direction does it point?
A.) Positive Skew, Right
Review
Any question about the exam?
Bring a calculator.
Good Luck Studying
